Remark 11  [CoralTDA for Power Filtration] Of course, after the previous result, the natural follow-up question is "Does CoralTDA also extends to power filtrations?". Unfortunately, the answer to this question is "No". The reason is by the nature of power filtration, one keeps adding edges to the existing vertices, and the degrees of the vertices increase in each step. This means \(k\) -cores significantly changes during this process. A simple counterexample can be found in [1]}, where the author completely classifies the topological types of clique complexes of the powers of cyclic graphs \(\lbrace C_n\rbrace \) , i.e., A graph which forms a circle. e.g., \(C_5\)  is a connected graph with 5 vertex and 5 edges. By definition, any 3-core of a cyclic graph is empty. However, Adamazsek's result show that for any dimension \(k\) , there exists topological features \(\widehat{PD}_k(C_n)\)  is nontrivial for \(n\ge 2k+3\) . If coralTDA could be extended to power filtrations, then \(\widehat{PD}_k(C_n)\)  would be trivial for any \(k\ge 3\) .

